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Let f:[0,oo)rarrR be a continuous functi...

Let `f:[0,oo)rarrR` be a continuous function such that `f(x)=1-2x+overset(x)underset(0)inte^(x-t)f(t)dt" for all "x in [0, oo).` Then, which of the following statements(s) is (are)) TRUE?

A

`(5)/(4)`

B

7

C

4

D

2

Text Solution

Verified by Experts

The correct Answer is:
C
Given , F `(x) int_(0)^(x) f (t) dt` By Leibnitz 's , rule , F ' (x) = f (x) . . . (i)
But `F(x^(2))=x^(2)(1+x)=x^(2)+x^(3)` [ given]
`rArr F(x)=x+x^(3//2)rArrF'(x) = 1+(3)/(2)x^(1//2)`
`rArrf(x)=F'(x)=1+(3)/(2)x^(1//2)` [ from Eq . (i)]
`rArr f(4)=1+(3)/(2)(4)^(1//2)rArrf(4)=1+(3)/(2)xx2=4`
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